Approximating $δ$-Covering

TL;DR

This work provides a comprehensive view of the approximability of $\delta$-Covering on unit-length graphs across all $\delta>0$. It shows a sharp threshold at $\delta=\tfrac{3}{2}$ where the problem transitions from constant-factor approximations to log-approximation hardness, and it leverages approximation-preserving reductions to Set Cover to obtain an $O(\log n)$ algorithm for the higher-$\delta$ regime. For $\delta<\tfrac{3}{2}$, it proves APX-hardness except when $\delta$ is a unit fraction, and it uses subdivision, translation, and wreath-product constructions to propagate hardness across subintervals, while also providing PU-based (UGC) lower bounds that tighten the achievable ratios near solvable points. The results illuminate how graph-structure and distance constraints interact to shape the approximability landscape, with implications for related continuous-discrete facility location problems and for understanding when efficient approximations are possible versus when hard-to-approximate regimes dominate.

Abstract

$δ$-Covering, for some covering range $δ>0$, is a continuous facility location problem on undirected graphs where all edges have unit length. The facilities may be positioned on the vertices as well as on the interior of the edges. The goal is to position as few facilities as possible such that every point on every edge has distance at most $δ$ to one of these facilities. For large $δ$, the problem is similar to dominating set, which is hard to approximate, while for small $δ$, say close to $1$, the problem is similar to vertex cover. In fact, as shown by Hartmann et al. [Math. Program. 22], $δ$-Covering for all unit-fractions $δ$ is polynomial time solvable, while for all other values of $δ$ the problem is NP-hard. We study the approximability of $δ$-Covering for every covering range $δ>0$. For $δ\geq 3/2$, the problem is log-APX-hard, and allows an $\mathcal O(\log n)$ approximation. For every $δ< 3/2$, there is a constant factor approximation of a minimum $δ$-cover (and the problem is APX-hard when $δ$ is not a unit-fraction). We further study the dependency of the approximation ratio on the covering range $δ< 3/2$. By providing several polynomial time approximation algorithms and lower bounds under the Unique Games Conjecture, we narrow the possible approximation ratio, especially for $δ$ close to the polynomial time solvable cases.

Figures (3)

• Figure 1: Upper bounds (as bold lines) and lower bounds under UGC (as thin lines) on the approximation ratio of $\delta\text{-Covering}$ plotted for $\delta \in (\tfrac{1}{2},1)$. The drawn intervals are half-open with the upper end excluded.
• Figure 2: Example for the reduction from Dominating Set to $\delta\text{-Covering}$ for $\delta \in [\frac{3}{2}, 2)$.
• Figure 3: Constructions (a), (b) and (c) for \ref{['theorem:uglbs']} for $x=1$ and input graph $K_3$. The dashed edges and vertices are the added gadgets, and the crosses mark the optimal placement of points on them. The thick edge segments are covered by those points. By the choice of $\delta$, for each construction $1 - \delta \leq \ell < \tfrac{1}{2}$. Thus a vertex cover of the original graph covers the remaining edge segments.

Theorems & Definitions (14)

• Lemma 1: Hartmann et al. DBLP:journals/mp/HartmannLW22
• Lemma 1
• Lemma 2: DBLP:journals/mp/HartmannLW22
• Lemma 2
• Lemma 2
• Lemma 2
• Lemma 2
• Definition 3: Subdivision Reduction
• Lemma 3
• Definition 4: Translation Reduction